US 7035782 B2 Abstract A method and apparatus are provided for solving a set of differential-algebraic equation arising in a circuit simulation is provided. A collocation method is applied to each differential-algebraic equation to discretize the set of differential-algebraic equations. A solution to the set of differential-algebraic equations based on the discretized differential-algebraic equation is then formed.
Claims(40) 1. A computer implemented method of simulating a circuit, the method comprising:
defining a set of differential-algebraic equations of the circuit;
defining a simulation time interval corresponding to the differential-algebraic equations;
dividing the simulation time interval into time intervals, wherein the time intervals have corresponding polynomials for each time interval, wherein each polynomial is a portion of an approximation to a desired solution of the differential-algebraic equations; and
applying a collocation method to discretize the differential-algebraic equations;
wherein:
the simulation time interval has M collocation points, and wherein M represents a highest degree of interpolating polynomials;
determining a smoothness for each interval, increasing an order of a solution for an interval if it is smooth, and splitting the interval into at least two sub-intervals if the interval is not smooth;
solving the differential-algebraic equations in each of the intervals wherein approximation to the desired solution of the differential-algebraic equations is
wherein
I
_{M }represents an M-point interpolation operator,u(t) represents a solution,
u
_{k}{tilde over ( )} represents Fourier coefficients, andT
_{k }(t) represents the interpolating polynomial.2. The method of
_{j }is a value of u(t_{j}) respectively, to be interpolated with polynomials.3. A computer implemented method of simulating a circuit, the method comprising:
defining a set of differential-algebraic equations of the circuit;
defining a simulation time interval corresponding to the differential algebraic equations;
dividing the simulation time interval into time intervals, wherein the time intervals have corresponding polynomials for each time interval, wherein each polynomial is a portion of an approximation to a desired solution of the differential-algebraic equations; and
applying a collocation method to discretize the differential-algebraic equations;
wherein:
the simulation time interval has M collocation points, and M represents highest a degree of interpolating polynomials;
determining a smoothness for each interval, increasing an order of a solution for an interval if it is smooth, and splitting the interval into at least two sub-intervals if the interval is not smooth;
solving the differential-algebraic equations in each of the intervals wherein approximation to the desired solution of the differential-algebraic equations is
wherein
I
_{M }represents an M-point interpolation operator,u(t) represents a solution,
u
_{k}{tilde over ( )} represents Fourier coefficients,T
_{k }(t) represents the interpolating polynomial; anda derivative of the approximation is
wherein
u
_{k}{tilde over ( )}′ represents the Fourier coefficients' derivative.4. The method of
u
_{k}{tilde over ( )}′ is computed fromu
_{k}{tilde over ( )}.5. The method of
6. The method of
7. A computer implemented method of solving a set of differential-algebraic equations arising in a circuit simulation, the method comprising:
applying a collocation method to each differential-algebraic equation to discretize the set of differential-algebraic equations;
forming a solution to the set of differential-algebraic equations based on the discretized differential-algebraic equations; and
determining an order of accuracy desired in intervals of the differential algebraic equations to be solved in the simulation;
wherein:
the set of differential-algebraic equations comprises a set of boundary-value differential-algebraic equations, and wherein the boundary-value differential-algebraic equations are discretized in the intervals, and wherein neighboring intervals share a boundary; and
the solution in a particular interval is smooth, and wherein the step of determining the order of accuracy desired in each interval comprises determining whether to increase the order of accuracy of the particular interval.
8. The method of
9. The method of
10. A computer implemented method of solving a set of differential-algebraic equations arising in a circuit simulation, the method comprising:
applying a collocation method to each differential-algebraic equation to discretize the set of differential-algebraic equations;
forming a solution to the set of differential-algebraic equations based on the discretized differential-algebraic equations; and
determining an order of accuracy desired in intervals of the differential algebraic equations to be solved in the simulation;
wherein:
the set of differential-algebraic equations comprises a set of boundary value differential-algebraic equations, and wherein the boundary-value differential-algebraic equations are discretized in the intervals, and wherein neighboring intervals share a boundary; and
the solution in a particular interval is not smooth, and wherein the step of determining the order of accuracy desired in each interval comprises splitting the particular interval into at least two subintervals.
11. The method of
12. The method of
13. The method of
14. The method of
15. The method of
solving the set of differential-algebraic equations using a Newton-Raphson iterative method; and
in each Newton-Raphson step of the Newton-Raphson iterative method, solving a linear Jacobian system using a linear iterative method.
16. The method of
17. The method of
18. A computer-readable medium carrying one or more sequences of one or more instructions for solving a set of differential-algebraic equations arising in a circuit simulation, the one or more sequences of one or more instructions including instructions which, when executed by one or more processors, cause the one or more processors to perform the steps of:
applying a collocation method to each differential-algebraic equation to discretize the set of differential-algebraic equations;
forming a solution to the set of differential-algebraic equations based on the discretized differential-algebraic equations; and
determining an order of accuracy desired in intervals of the differential algebraic equations to be solved in the simulation;
wherein:
the set of differential-algebraic equations comprises a set of boundary value differential-algebraic equations, and wherein the boundary-value differential-algebraic equations are discretized in the intervals, and wherein neighboring intervals share a boundary; and
the solution in a particular interval is not smooth, and wherein the step of determining the order of accuracy desired in each interval further causes the processors to carry out the step of splitting the particular interval into at least two subintervals.
19. The computer-readable medium of
20. The computer-readable medium of
21. The computer-readable medium of
22. The computer-readable medium of
23. The computer-readable medium of
24. The computer-readable medium of
25. The computer-readable medium of
solving the set of differential-algebraic equations using a Newton-Raphson iterative method; and
in each Newton-Raphson step of the Newton-Raphson iterative method, solving a linear Jacobian system using a linear iterative method.
26. The computer-readable medium of
27. The computer-readable medium of
28. A computer-readable medium carrying one or more sequences of one or more instructions for solving a set of differential-algebraic equations arising in a circuit simulation, the one or more sequences of one or more instructions including instructions which, when executed by one or more processors, cause the one or more processors to perform the steps of:
wherein:
the set of differential-algebraic equations comprises a set of boundary value differential-algebraic equations, and wherein the boundary-value differential-algebraic equations are discretized in the intervals, and wherein neighboring intervals share a boundary; and
the solution in a particular interval is smooth, and wherein the step of determining the order of accuracy desired in each interval further causes the processor to carry out the step of determining whether to increase the order of accuracy of the particular interval.
29. A computer implemented method of simulating an RF circuit, comprising the steps of:
determining a plurality of differential-algebraic equations describing operation of the RF circuit;
determining a set of Chebyshev Gauss-Lobatto collocation points for the plurality of differential-algebraic equations, producing a set of intervals;
discretizing each of the differential-algebraic equations based on the Chebyshev Gauss-Lobatto collocation point intervals;
determining a smoothness for each interval, increasing an order of a solution for an interval if it is smooth, and splitting the interval into at least two sub-intervals if the interval is not smooth;
solving the differential-algebraic equations in each of the intervals; and
simulating the RF circuit based on the solved intervals.
30. The method according to
enforcing continuity of the solution at each interval boundary; and
enforcing a periodic boundary condition at each first and last interval boundaries.
31. A computer implemented method of simulating an RF circuit, comprising the steps of:
determining a plurality of differential-algebraic equations describing operation of the RF circuit;
determining a set of Chebyshev Gauss-Lobatto collocation points for the plurality of differential-algebraic equations, producing a set of intervals;
discretizing each of the differential-algebraic equations based on the Chebyshev Gauss-Lobatto collocation point intervals;
solving the differential-algebraic equations in each of the intervals; and
simulating the RF circuit based on the solved intervals;
wherein the step of solving comprises applying a set of at least one high order solution to at least one of the intervals and applying at least one solution from a set of low order solutions to a plurality of the intervals.
32. The method according to
33. The method according to
dividing the intervals into smooth and non-smooth intervals, applying higher order solutions to the smooth intervals, and applying lower order solutions to the non-smooth intervals.
34. A computer implemented method of simulating an RF circuit, comprising the steps of:
determining a plurality of differential-algebraic equations describing operation of the RF circuit;
determining a set of Chebyshev Gauss-Lobatto collocation points for the plurality of differential-algebraic equations, producing a set of intervals;
discretizing each of the differential-algebraic equations based on the Chebyshev Gauss-Lobatto collocation point intervals;
solving the differential-algebraic equations in each of the intervals; and
simulating the RF circuit based on the solved intervals;
wherein the Chebyshev Gauss-Lobatto collocation points produce a small number of intervals in areas in which the differential-algebraic equations exhibit high convergence, and a large number of intervals in areas where the differential-algebraic equations exhibit low convergence.
35. The method according to
36. A computer implemented method of simulating a circuit, comprising the steps of:
determining a plurality of differential-algebraic equations describing operation of the circuit;
determining a set of collocation points for the plurality of differential-algebraic equations, producing a set of intervals comprising at least one high convergence interval and a plurality of low convergence intervals;
applying a higher order solution in the at least one high convergence interval;
applying a lower order solution in the low convergence intervals; and
simulating the circuit response using the higher and lower order solutions.
37. The method according to
38. The method according to
39. The method according to
enforcing continuity of the solutions at each interval boundary; and
enforcing a periodic boundary condition at at least one of the first and last interval boundaries.
40. The method according to
Description This application claims the benefit of Provisional Application No. 60/208,724, filed Jun. 2, 2000. A portion of the disclosure of this patent document contains material which is subject to copyright protection. The copyright owner has no objection to the facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights whatsoever. 1. Field of Invention The present invention relates generally to analyzing integrated circuits and, more particularly, to techniques for performing simulations of radio frequency (RF) integrated circuits. 2. Discussion of Background The exploding demand for high performance wireless products has increased the need for efficient and accurate simulation techniques for RF integrated circuits. RF circuit simulation is difficult because RF circuits typically contain signals with multiple-timescale properties, as usually the data and carrier signals in a system are separated in frequency by several orders of magnitude. Special-purpose RF simulators exploit the sparsity of the spectrum in order to make the computations tractable. See K. S. Kundert, “Introduction to RF Simulation and Its Applications,” IEEE J. Sol. State Circuits, September 1999. For example, instead of performing a long transient analysis of a circuit driven by a periodic source, we may seek to find the steady-state directly. Unfortunately, periodic-steady-state computation may be problematic in that it can be inefficient and inaccurate using conventional methods. Two numerical methods commonly used in steady-state computation are the shooting-Newton method, based on low-order finite difference discretizations such as the second-order Gear method (see K. S. Kundert, “Introduction to RF Simulation and Its Applications,” IEEE J. Sol. State Circuits, September 1999), and the harmonic balance method, based on high-order spectral discretizations (see “Nonlinear Circuit Analysis Using The Method of Harmonic Balance-A Review of The Art,” Part I-Introductory Concepts, Int. J. Microwave and Millimeter Wave Computer Aided Engineering, vol. 1, no. 1, 1991). An advantage of the low-order polynomial-based methods is that these methods can select time-points based on localized error estimates and as a result can easily handle sharp transitions in circuit waveforms. The harmonic balance method, on the other hand, has the advantage of attaining spectral accuracy for smooth waveforms. Recent developments in matrix-free Krylov-subspace algorithms have made these methods even more popular as they can be used to analyze circuits with thousands of devices. For more information on matrix-free Kylov-subspace algorithms, see R. Telichevesky, K. Kundert and J. K. White, “Efficient AC and Noise Analysis of Two-Tone RF Circuits,” Proc. 33rd Design Automation Conference, June, 1996; D. Long and R. Melville and K. Ashby and B. Horton, “Full Chip Harmonic Balance,” Proc. Custom Integrated Circuits Conference, May, 1997. High precision computations are often necessary in RF circuit simulation. For example, accurately computing the noise figure of a highly nonlinear RF circuit often requires accurate determination of the periodic operating point up to many multiples of the fundamental frequency, as noise may be translated from very high frequencies by the mixing action of the time-varying circuit elements to appear at the output. High accuracy is easy to achieve for smooth waveforms using spectral approximation in the harmonic balance method. However, non-linear circuits often produce waveforms that have sharp-transition regions. The waveforms may only be C To help handle sharp transitions, Nastov and White introduced the time-mapped harmonic balance (TMHB) method. See O. J. Nastov and J. K. White, “Time-Mapped Harmonic Balance,” Proc. 36th Design Automation Conference, New Orleans, La., June, 1999. In the TMHB method, a time-map function is used to map the solution in the original time space to a space where the solution is more smooth. This method is an improvement over traditional harmonic balance, but is still restricted, since, theoretically speaking, a C The low-order multi-step methods, such as the Gear methods, typically used in the shooting method offer excellent flexibility in locally adapting the timesteps. For low to moderate accuracy, these methods are preferred for problems that are highly nonlinear and/or contain sharp transitions. However, low-order multi-step methods are not efficient at high precision because very fine discretizations are needed, resulting in loss of speed and increased memory requirements. In addition, the multistep discretizations used in the shooting method must be causal, and so have stability problems associated with using one-side approximations, namely, performing differentiation using backward differences. It is generally accepted that multistep methods of order greater then three or four are not sufficiently stable, and in practice even the third order methods have relatively stringent stability restrictions on how rapidly timesteps may be varied. As a result, multi-step methods of order higher than 2 are not widely used in circuit simulation. Often, because of the backward-looking nature of the approximation, it is necessary to decrease the order after sharp-transition points or C In sum, most RF circuit analysis tools use either shooting-Newton or harmonic balance methods. Unfortunately, neither method can efficiently achieve high accuracy on strongly nonlinear circuits possessing waveforms with rapid transitions. It has been recognized that what is needed is a method for performing efficient, highly accurate radio frequency circuit simulation. Broadly speaking, the present invention fills these needs by providing a method and device for multi-interval Chebyshev collocation. It should be appreciated that the present invention can be implemented in numerous ways, including as a process, an apparatus, a system, a device or a method. Several inventive embodiments of the present invention are described below. The present invention provides a multi-interval-Chebyshev (MIC) method for solving boundary-value differential-algebraic equations arising in radio frequency (RF) circuit simulation. The varying-order, varying-timestep MIC scheme effectively adapts to different types of waveforms that appear in circuit simulation. The method has spectral accuracy for smooth wave-forms. In some embodiments, the method is efficient in resolving sharp transitions and nonlinear effects with high accuracy. Note that purely low-order schemes have difficulty achieving accuracy at reasonable cost, and high-order methods such as harmonic balance lose accuracy due to oscillations in sharp-transition regions. Preconditioning techniques are presented herein that make the cost of the present scheme comparable to backward differential formula-based (BDF-based) shooting-Newton methods. In one embodiment, a method of simulating a circuit is provided. The method comprises the following: defining a differential-algebraic equation of the circuit; defining a simulation time interval corresponding to the differential-algebraic equation; dividing the simulation time interval into time intervals, wherein the time intervals have corresponding polynomials for each time interval, wherein each polynomial is a portion of an approximation to a desired solution of the differential-algebraic equation; and applying a collocation method to discretize the differential-algebraic equation. In another embodiment, a method of solving a set of differential-algebraic equation arising in a circuit simulation is provided. The method comprises the following: applying a collocation method to each differential-algebraic equation to discretize the set of differential-algebraic equations; and forming a solution to the set of differential-algebraic equations based on the discretized differential-algebraic equation. Advantageously, the present invention provides a method for efficient, high-accuracy radio frequency (RF) circuit simulation. Because of inevitable discontinuities in the solution waveforms, residuals, or device equations, the discretization is performed by breaking the simulation interval, of length T, into m timestep intervals of length h The present invention will be readily understood by the following detailed description in conjunction with the accompanying drawings. To facilitate this description, like reference numerals designate like structural elements. An invention is disclosed for a method and device for multi-interval Chebyshev collocation for efficient, highly accurate radio frequency circuit simulation. Numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be understood, however, to one skilled in the art, that the present invention may be practiced without some or all of these specific details. General Overview of Circuit Equation Discretization Circuit behavior is usually described by a set of N nonlinear differential-algebraic equations (DAEs) that can be written as We now contrast three ways of discretizing the time-derivative operator: linear multistep methods, harmonic balance methods, and Chebyshev methods. As representative of linear multistep methods we consider the Gear methods. Given the values, u(t The Gear methods are implicit in only “one-point-at-a-time”. They are called one-stage methods. At each timepoint, for a method of order p, the solution vector at only p+1 points is involved, and if p previous timepoints are known, a nonlinear system of size N must be solved to find the solution at the next timepoint. Each step of a shooting method involves the factorization of the circuit Jacobian matrix. Because this matrix is usually very sparse, it can be factored with few fill-ins, in nearly O(N) time. The order of polynomial approximation p is usually fixed at a low value, and increased accuracy is obtained by locally changing the timesteps based on estimates of the truncation errors induced by the order-p polynomial approximation. Note that higher-order Gear methods can not be used at points after C For contrast, consider a particular (time-domain) construction of the harmonic balance method, known as Fourier-collocation. We seek to form the solution u(t) by an interpolating trigonometric polynomial of degree M:
The differentiation procedure may be performed efficiently to high order by noting that in the frequency domain, the differentiation operator becomes a multiplication operator, so by using the FFT to convert the u(t Accuracy in the harmonic balance method is usually achieved by increasing the order of approximation. For a given set of collocation points, the maximal possible order of approximation is used. With C Chebyshev Polynomials A preferred class of basis functions that also possess good approximation properties, including the possibility of spectral convergence property, are the Chebyshev polynomials. However, Chebyshev polynomials have the advantage that they can be applied when the simulation domain has been decomposed into multiple intervals in order to resolve complex or singular behavior, as has been done for electromagnetic wave problems in complex geometries. For more information on electromagnetic wave problems in complex geometries, see B. Yang, D. Gottlieb and J. S. Hesthaven, “Spectral Simulations of Electromagnetic Wave Scattering,” J. Comput. Phys., vol. 135, pp. 216–230, 1997. The following discussion provides the basics of the Chebyshev collocation method. Given the values, u(t We can also obtain a derivative matrix D by noting that the interpolating polynomial can be expressed in terms of a Lagrange inter-polation polynomial g The entries of the matrix are An important aspect of the multi-interval Chebyshev (MIC) approach is to break the simulation domain [0, T] into multiple intervals, [0, T First, as in harmonic balance, high accuracy is achievable because in each interval all the desirable approximation properties of Chebyshev polynomials can be exploited. Chebyshev polynomials are usually close to the optimal polynomial approximation of a given order, and spectral convergence can be achieved if order-based (p-type) refinement is used in each interval. Because of the rapid convergence, once a proper set of intervals is chosen high accuracy can be obtained at little marginal cost. Conversely, there will be few intervals in regions where the solution is smooth, because the intervals there can be quite large. Second, as in the lower-order finite difference schemes, the timestep intervals can be adaptively chosen to resolve nonlinear or abrupt transition behavior. The impact of nonlinear behavior can some-times be difficult to diagnose. Solution waveforms, such as voltages, may be quite smooth, but residuals, such as currents, will be much more difficult to represent with a Fourier series if the circuit equations are very nonlinear. In the MIC method, if the intervals are chosen such that discontinuities occur only at the interval boundaries, then the discontinuities can be handled efficiently. Around a transition or strongly nonlinear region, more intervals can be placed in a denser timestep spacing, and possibly the order of the method lowered. Third, the method has excellent numerical stability properties. It can be shown that the scheme of the present invention is a member of a particular class of implicit Runge-Kutta (IRK) methods, the collocation-IRK methods. For more information on solving differential equations, see E. Hairer and G. Wanner, “Solving Ordinary Differential Equations II,” Springer-Verlag, 1991. Higher-order implicit Runge-Kutta methods are multi-stage methods, that is, they are implicit in more than one point at a time. Because of the multiple implicit states, and unlike multi-step methods such as the Gear methods, IRK methods can be constructed that are A-stable even at high order. A-stable means a scheme that is numerically stable for systems with poles located anywhere in the left-half-plane, including the entire imaginary axis. The particular schemes used in this discussion are not A-stable, but it has been proved that the schemes of all orders used here are what is called A(α)-stable, meaning that if λ is a pole of the system being integrated, then the numerical scheme is stable with timestep h for all z=hλ in the left-half-plane, except possibly within an angle α of the imaginary axis. For the methods of the present invention, α is usually less than one degree, and the stability region only excludes a small semi-circle near the imaginary axis, which can be avoided by appropriate choice of timesteps. In addition, because the Gauss-Lobatto-Chebyshev points are preferred for collocation (implying a collocation point is always placed at the forward interval boundary), the particular construction here gives methods that are L- or stiffly stable, just as the Gear methods are. Accordingly, the method is suitable for solving differential-algebraic equations as initial value problems. This property allows the use of shooting techniques to solve boundary value problems or construct preconditioners. In fact, in a certain sense that is important in practical applications, the MIC approach is more stable than the Gear methods. Since unlike the Gear methods, the MIC scheme retains excellent stability properties even when the timesteps are rapidly varied, higher-order schemes can be used near sharp transition points. Finally, the discretization scheme is robust. Unlike harmonic balance schemes, where sufficiently sharp transitions can induce oscillations in the global basis that are next to impossible to eliminate, in the worst possible case the discretization in a single MIC interval degenerates to the backward-Euler scheme which is robust and well-tested. In practice, the need to drop below the second order scheme is rare. Efficient Implementation of Chebyshev Collocation Methods For simplicity of presentation, the following discussion involves Chebyshev-collocation methods for the case of two intervals: I
At each Newton iteration step, we must solve a linear system
In each interval a “local” preconditioning approach is preferred that is motivated by techniques used for large-scale harmonic balance analysis. But because [0, T] is decomposed into multiple intervals, a preconditioner can be devised that is much more efficient than the corresponding harmonic balance preconditioner. In this way, the “global” preconditioning approach is similar to that used in R. Telichevesky, K. Kundert and J. K. White, “Efficient AC and Noise Analysis of Two-Tone RF Circuits,” Proc. 33rd Design Automation Conference, June, 1996, for shooting methods, where a preconditioner is obtained for finite difference equations by dropping the top-right corner entries in the left-hand-side matrix of the linear system. In harmonic balance, it is common to construct a preconditioner by assuming the capacitance and conductance matrices C, G respectively can be well represented by (piecewise) constant approximations. Both these techniques are adopted in the present invention. For example, a single {tilde over (C)} is used as an approximation of all the capacitance matrices in interval I The advantage of this preconditioner over harmonic balance preconditioners is that one can approximate the capacitance and conductance matrices with averages separately in each interval at little computational cost, in contrast to using global averages which make the preconditioner very inefficient in simulations of nonlinear circuits. Because the intervals are chosen to resolve the nonlinear behavior, we can expect the cost of applying the preconditioner to be relatively independent of the degree of nonlinearity of the circuit. If the circuit is very nonlinear, there will be very many intervals, yet precisely for this reason the preconditioner will still be a good approximation to the original matrix, and in fact will be nearly as cheap to apply as the shooting-method preconditioner. In a practical implementation directly solving the MIC equations is not preferred. Instead, a low-order shooting-Newton method is used to obtain an initial approximation to the solution waveform. For steady-state problems, the shooting-Newton method has a good global convergence rate in periodic steady state (PSS) simulations and can rapidly reach the neighborhood of the steady state. From the result of the low-order solution method, the time interval [0, T] is decomposed into a number of subintervals. The objective of this first decomposition is to avoid sharp transitions and C Once an initial solution is obtained, an adaptive refinement scheme is used to split the intervals in any additional sharp-transition regions that may appear and increase the order of approximation in other regions where accuracy is not satisfactory. In this stage, both h-type (interval decomposition) and p-type (order increase) refinement are used to achieve the required accuracy in the most efficient way. Error control can be performed by examining the trailing Chebyshev coefficients as indicators of potential numerical truncation error. Initial experiments have been conducted using the following simple error criteria, In this section, the spectral accuracy of the multidomain Chebyshev collocation method in periodic steady state computations is provided. Efficiency of using h-type (domain decomposition) and p-type (polynomial order increase) refinement in resolving waveforms with sharp transitions is shown. The method's ability to resolve sharp transitions with high accuracy is verified. The efficiency of the preconditioning technique is also demonstrated by the solution of the linear system using the matrix-free Krylov-subspace approach. In the first example, the spectral accuracy of the multidomain Chebyshev collocation methods is shown by computing the periodic steady-state solution of a crystal filter. In the second example, the efficiency of using both h-type and p-type refinement in resolving solutions with sharp transitions is provided. Nonlinearity is introduced into the previous filter example by placing a mixer before it. Three refinement strategies are compared: h-type refinement only, p-type refinement only, and hp-type (h-type and p-type) refinement. In the last example, the periodic steady-state of a very nonlinear DC—DC converter is calculated, similar to a circuit analyzed in O. J. Nastov and J. K. White, “Time-Mapped Harmonic Balance,” Proc. 36th Design Automation Conference, New Orleans, La., June, 1999. The final mesh of the proposed approach is obtained through three steps of hp-refinement. By using both h-type and p-type refinement, the method selects high-order approximations, with large intervals, in the smooth region and low-order approximations, with small intervals, in the sharp-transition region.
The second order Gear method requires two orders of magnitude more timepoints to reach comparable accuracy, while using global high-order basis functions, such as used in harmonic balance, leads to large errors. These large errors can be explained by examining the strictly high-order solution shown in It is difficult to make precise comparisons of the efficiency of the method, because the shooting solution is used to achieve the initial mesh refinement, and because the cost varies depending on how many steps of mesh refinement are needed, the choice of approximation order in each subinterval, etc. However, the method of the present invention generally only uses 100–150% of the computation time of the initial low-order shooting-Newton guess to attain an accuracy of orders of magnitude higher. The GMRES iterative solver requires somewhat more GMRES iterations (usually between about 1 and 3 factors) per Newton iteration than the low-order shooting-Newton method. However, few Newton iterations beyond the shooting stage are typically required to reach convergence. In the DC—DC converter example, the number of GMRES iterations required is only around 10, compared with hundreds or even thousands of iterations required by the harmonic balance method in O. J. Nastov and J. K. White, “Time-Mapped Harmonic Balance,” Proc. 36th Design Automation Conference, New Orleans, La., June, 1999. More importantly, because many fewer timepoints are used to achieve high accuracy than other methods, the memory usage of our scheme, which is usually the limiting factor in determining capacity of an RF simulator, is potentially very much lower. As a study of efficiency on a larger example, what is used is the benchmark receiver circuit from R. Telichevesky, K. Kundert and J. K. White, “Efficient AC and Noise Analysis of Two-Tone RF Circuits,” Proc. 33rd Design Automation Conference, June, 1996. With the accuracy requirement at a fixed high precision (four digits in the fourth harmonic), the MIC method has shown a performance improvement over the low-order shooting-based method of 13× in CPU time and 20×in timepoint usage (proportional to memory). If instead the number of timepoints is fixed at a similar number in both methods, an additional 150% of the time of an initial low-accuracy shooting solution may be needed by the MIC method to increase the accuracy by 300×. Therefore, in one embodiment, the present invention may be practiced as a method of simulating a circuit, the method comprising steps for defining a differential-algebraic equation of the circuit, defining a simulation time interval corresponding to the differential-algebraic equation, and dividing the simulation time interval into time intervals. The time intervals may include corresponding polynomials for each time interval, wherein each polynomial is a portion of an approximation to a desired solution of the differential-algebraic equation, and the method includes the step of applying a collocation method to discretize the differential-algebraic equation. In the method, the simulation time interval has collocation points, and wherein the interpolating polynomial has a degree of M, and the approximation to the desired solution of the differential-algebraic equations is The invention may also be described as a method of simulating a circuit, the method comprising the steps of defining a differential-algebraic equation of the circuit, defining a simulation time interval corresponding to the differential-algebraic equation, dividing the simulation time interval into time intervals, wherein the time intervals have corresponding polynomials for each time interval, wherein each polynomial is a portion of an approximation to a desired solution of the differential-algebraic equation, and applying a collocation method to discretize the differential-algebraic equation. In the method, the simulation time interval has collocation points, and wherein the interpolating polynomial has a degree of M, the approximation to the desired solution of the differential-algebraic equations is
In another embodiment, the present invention is a method of solving a set of differential-algebraic equations arising in a circuit simulation, the method comprising the steps of, applying a collocation method to each differential-algebraic equation to discretize the set of differential-algebraic equations; forming a solution to the set of differential-algebraic equations based on the discretized differential-algebraic equation, and determining an order of accuracy desired in each interval, wherein, the set of differential-algebraic equations comprises a set of boundary-value differential-algebraic equations, and wherein the boundary-value differential-algebraic equations are discretized in intervals, and wherein neighboring intervals share a boundary, and the solution in a particular interval is not smooth, and wherein the step of determining the order of accuracy desired in each interval comprises splitting the particular interval into at least two subintervals. The method may include that the set of differential-algebraic equations comprises a set of boundary-value differential-algebraic equations, and the boundary-value differential-algebraic equations include a first and a last interval. In yet another embodiment, the present invention is a computer-readable medium carrying one or more sequences of one or more instructions for solving a set of differential-algebraic equations arising in a circuit simulation, the one or more sequences of one or more instructions including instructions which, when executed by one or more processors, cause the one or more processors to perform the steps of, applying a collocation method to each differential-algebraic equation to discretize the set of differential-algebraic equations, and forming a solution to the set of differential-algebraic equations based on the discretized differential-algebraic equation, wherein, the set of differential-algebraic equations comprises a set of boundary-value differential-algebraic equations, the boundary-value differential-algebraic equations are discretized in intervals, and wherein neighboring intervals share a boundary, the instructions further cause the processor to carry out the step of determining an order of accuracy desired in each interval, and the solution in a particular interval is not smooth, and wherein the step of determining the order of accuracy desired in each interval further causes the processor to carry out the step of splitting the particular interval into at least two subintervals. The computer-readable medium may include that the set of differential-algebraic equations comprises a set of boundary-value differential-algebraic equations, and the boundary-value differential-algebraic equations include a first and a last interval. System and Method Implementation Portions of the present invention may be conveniently implemented using a conventional general purpose or a specialized digital computer or microprocessor programmed according to the teachings of the present disclosure, as will be apparent to those skilled in the computer art. Appropriate software coding can readily be prepared by skilled programmers based on the teachings of the present disclosure, as will be apparent to those skilled in the software art. The invention may also be implemented by the preparation of application specific integrated circuits (ASIC's) or by interconnecting an appropriate network of conventional component circuits, as will be readily apparent to those skilled in the art. The present invention includes a computer program product which is a storage medium (media) having instructions stored thereon/in which can be used to control, or cause, a computer to perform any of the processes of the present invention. The storage medium can include, but is not limited to, any type of disk including floppy disks, mini disks (MD's), optical discs, DVD, CD-ROMS, micro-drive, and magneto-optical disks, ROMs, RAMs, EPROMs, EEPROMs, DRAMs, VRAMs, flash memory devices (including flash cards), magnetic or optical cards, nanosystems (including molecular memory ICs), RAID devices, remote data storage/archive/warehousing, or any type of media or device suitable for storing instructions and/or data. Stored on any one of the computer readable medium (media), the present invention includes software for controlling both the hardware of the general purpose/specialized computer or microprocessor, and for enabling the computer or microprocessor to interact with a human user or other mechanism utilizing the results of the present invention. Such software may include, but is not limited to, device drivers, operating systems, and user applications. Ultimately, such computer readable media further includes software for performing the present invention, as described above. Included in the programming (software) of the general/specialized computer or microprocessor are software modules for implementing the teachings of the present invention, including, but not limited to, applying Chebyshev collocation to boundary-value differential equations to discretize a set of BVD equations, and forming a solution to the set of BVD equations based on the discretized BVD equation, according to processes of the present invention. In the foregoing specification, the invention has been described with reference to specific embodiments thereof. It will, however, be evident that various modifications and changes may be made thereto without departing from the broader spirit and scope of the invention. The specification and drawings are, accordingly, to be regarded in an illustrative rather than a restrictive sense. Patent Citations
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